Number / Algebra / Functions / Geometry
The real number system
Extend the properties of exponents
NC.M1.N-RN.2
Algebra
Overview
Seeing structure in expressions
Interpret the structure of expressions
NC.M1.A-SSE.1a
NC.M1.A-SSE.1b
Write expressions in equivalent forms to solve problems
NC.M1.A-SSE.3
Perform arithmetic operations on polynomials
Perform arithmetic operations on polynomials
NC.M1.A-APR.1
Understand the relationship between zeros and factors of polynomials
NC.M1.A-APR.3
Creating Equations
Create equations that describe numbers or relationships
NC.M1.A-CED.1
NC.M1.A-CED.2
NC.M1.A-CED.3
NC.M1.A-CED.4 / Reasoning with equations and inequalities
Understand solving equations as a process of reasoning and explain the reasoning
NC.M1.A-REI.1
Solve equations and inequalities in one variable
NC.M1.A-REI.3
NC.M1.A-REI.4
Solve systems of equations
NC.M1.A-REI.5
NC.M1.A-REI.6
Represent and solve equations and inequalities graphically
NC.M1.A-REI.10
NC.M1.A-REI.11
NC.M1.A-REI.12 / Overview
Interpreting Functions
Understand the concept of a function and use function notation
NC.M1.F-IF.1
NC.M1.F-IF.2
NC.M1.F-IF.3
Interpret functions that arise in applications in terms of a context
NC.M1.F-IF.4
NC.M1.F-IF.5
NC.M1.F-IF.6
Analyze functions using different representations
NC.M1.F-IF.7
NC.M1.F-IF.8a
NC.M1.F-IF.8b
NC.M1.F-IF.9
Building Functions
Build a function that models a relationship between two quantities
NC.M1.F-BF.1a
NC.M1.F-BF.1b
NC.M1.F-BF.2
Linear, Quadratics and Exponential Models
Construct and compare linear and exponential models to solve problems
NC.M1.F-LE.1
NC.M1.F-LE.3
Interpret expressions for functions in terms of the situations they model
NC.M1.F-LE.5 / Overview
Expressing geometric properties with equations
Use coordinates to prove simple geometric theorems algebraically
NC.M1.G-GPE.4
NC.M1.G-GPE.5
NC.M1.G-GPE.6
Statistics & Probability
Overview
Interpreting Categorical and Quantitative Data
Summarize, represent, and interpret data on a single count or measurement variable
NC.M1.S-ID.1
NC.M1.S-ID.2
NC.M1.S-ID.3
Summarize, represent, and interpret data on two categorical and quantitative variables
NC.M1.S-ID.6a
NC.M1.S-ID.6b
NC.M1.S-ID.6c
Interpret linear models
NC.M1.S-ID.7
NC.M1.S-ID.8
NC.M1.S-ID.9
Number – The Real Number System
NC.M1.N-RN.2Extend the properties of exponents.
Rewrite algebraic expressions with integer exponents using the properties of exponents.
Concepts and Skills / The Standards for Mathematical Practices
Pre-requisite / Connections
- Using the properties of exponents to create equivalent numerical expressions (8.EE.1)
7 – Look for and make use of structure
8 – Look for and express regularity in repeated reasoning
Connections / Disciplinary Literacy
- Use operations to rewrite polynomial expressions (NC.M1.A-APR.1)
Vocabulary – base, exponent, index
Students should be able to justify their steps in rewriting algebraic expressions.
Mastering the Standard
Comprehending the Standard / Assessing for Understanding
Students extend the properties of integer exponents learned in middle schoolwith numerical expressions to algebraic expressions.
The process of “simplifying square roots” is not an expectation for Math 1 studentsIn Math 2, students will extend the properties of exponents to rational exponentsand rewrite, “simplify” all square roots. / Students should be able to use the properties of exponents to write expression into equivalent forms.
Example: Rewrite the following with positive exponents:
a)
b)
Students should be able to use the new skills of applying the properties of exponents with skills learned in previous courses.
Example: Simplify:
In 8th grade, students learned to evaluate the square roots of perfect squares and the cube root of perfect cubes. In Math 1, students can combine this previous skill with algebraic expressions. When addressing a problem like this in Math 1, students should be taught to rewrite the expression using the properties of exponents and then using inverse operations to rewrite. For example, .
In Math 1, the limitation from 8th grade of evaluating square roots of perfect squares and cube root of perfect cubes still applies.
Instructional Resources
Tasks / Additional Resources
Raising to the Zero and Negative Power (Illustrative Mathematics) NEW
Back to: Table of Contents
Algebra, Functions & Function FamiliesNC Math 1 / NC Math 2 / NC Math 3
Functions represented as graphs, tables or verbal descriptions in context
Focus on comparing properties of linear function to specific non-linear functions and rate of change.
•Linear
•Exponential
•Quadratic / Focus on properties of quadratic functions and an introduction to inverse functions through the inverse relationship between quadratic and square root functions.
•Quadratic
•Square Root
•Inverse Variation / A focus on more complex functions
•Exponential
•Logarithm
•Rational functions w/ linear denominator
•Polynomial w/ degree three
•Absolute Value and Piecewise
•Intro to Trigonometric Functions
A Progression of Learning of Functions through Algebraic Reasoning
The conceptual categories of Algebra and Functions are inter-related. Functions describe situations in which one quantity varies with another. The difference between the Function standards and the Algebra standards is that the Function standards focus more on the characteristics of functions (e.g. domain/range or max/min points), function definition, etc. whereas the Algebra standards provide the computational tools and understandings that students need to explore specific instances of functions. As students progress through high school, the coursework with specific families of functions and algebraic manipulation evolve. Rewriting algebraic expressions to create equivalent expressions relates to how the symbolic representation can be manipulated to reveal features of the graphical representation of a function.
Note: The Numbers conceptual category also relates to the Algebra and Functions conceptual categories. As students become more fluent with their work within particular function families, they explore more of the number system. For example, as students continue the study of quadratic equations and functions in Math 2, they begin to explore the complex solutions. Additionally, algebraic manipulation within the real number system is an important skill to creating equivalent expressions from existing functions.
Back to: Table of Contents
Algebra – Seeing Structure in Expressions
NC.M1.A-SSE.1aInterpret the structure of expressions.
Interpretexpressions thatrepresent aquantityintermsof itscontext.
- Identifyandinterpretpartsofa linear,exponential,or quadraticexpression,includingterms, factors,coefficients, andexponents.
Concepts and Skills / The Standards for Mathematical Practices
Pre-requisite / Connections
- Identify parts of an expression using precise vocabulary (6.EE.2b)
- Interpret numerical expressions written in scientific notation (8.EE.4)
- For linear and constant terms in functions, interpret the rate of change and the initial value (8.F.4)
2 – Reason abstractly and quantitatively.
4 – Model with mathematics
7 – Look for and make use of structure.
Connections / Disciplinary Literacy
- Creating one and two variable equations (NC.M1.A-CED.1, NC.M1.A-CED.2, NC.M1.A-CED.3)
- Interpreting part of a function to a context (NC.M1.F-IF.2, NC.M1.F-IF.4, NC.M1.F-IF5, NC.M1.F-IF.7, NC.M1.F-IF.9)
- Interpreting changes in the parameters of a linear and exponential function in context (NC.M1.F-LE.5)
New Vocabulary: Quadratic term, exponential term
Mastering the Standard
Comprehending the Standard / Assessing for Understanding
The set of A-SSE standards requires students:
- to write expressions in equivalent forms to reveal key quantities in terms of its context.
- to choose and use appropriate mathematics to analyze situations.
Students are expected to recognize the parts of a quadratic expression, such as the quadratic, linear and constant term, or factors.
For exponential expressions, students should recognize factors, the base, and exponent(s).
Students extend beyond simplifyingtointerpretthe components ofan algebraic expression. / Students should recognize that in the expression , “2” is the coefficient, “2” and “x” are factors, and “1” is a constant, as well as “2x” and “1” being terms of the binomial expression. Also, a student recognizes that in the expression , 4 is the coefficient, 3 is the factor, and x is the exponent. Development and proper use of mathematical language is an important building block for future content. Using real-world context examples, the nature of algebraic expressions can be explored.
Example: The height (in feet) of a balloon filled with helium can be expressed by where s is the number of seconds since the balloon was released. Identify and interpret the terms and coefficients of the expression.
Example:The expression describes the height in meters of a basketball t seconds after it has been thrown vertically into the air. Interpret the terms and coefficients of the expression in the context of this situation.
Example: The expression describes the cost of a new car years after it has been purchased. Interpret the terms and coefficients of the expression in the context of this situation.
Instructional Resources
Tasks / Additional Resources
Delivery Trucks (Illustrative Mathematics) / Interpreting Algebraic Expressions (Mathematics Assessment Project – FAL)
Back to: Table of Contents
Algebra – Seeing Structure in Expressions
NC.M1.A-SSE.1bInterpretthe structure ofexpressions.
Interpretexpressions thatrepresent aquantityintermsof itscontext.
- Interpret a linear,exponential,orquadraticexpression made of multiplepartsas acombination of entitiesto give meaningtoanexpression.
Concepts and Skills / The Standards for Mathematical Practices
Pre-requisite / Connections
- Interpret a sum, difference, product, and quotient as a both a whole and as a composition of parts (6.EE.2b)
- Understand that rewriting expressions into equivalent forms can reveal other relationships between quantities (7.EE.2)
- Interpret numerical expressions written in scientific notation (8.EE.4)
2 – Reason abstractly and quantitatively.
4 – Model with mathematics
7 – Look for and make use of structure.
Connections / Disciplinary Literacy
- Factor to reveal the zeros of functions and solutions to quadratic equations (NC.M1.A.SSE.3)
- Creating one and two variable equations (NC.M1.A-CED.1, NC.M1.A-CED.2, NC.M1.A-CED.3)
- Interpreting part of a function to a context (NC.M1.F-IF.2, NC.M1.F-IF.4, NC.M1.F-IF5, NC.M1.F-IF.7, NC.M1.F-IF.9)
- Interpreting changes in the parameters of a linear and exponential function in context (NC.M1.F-LE.5)
New Vocabulary: exponential expression, quadratic expression
Mastering the Standard
Comprehending the Standard / Assessing for Understanding
The set of A-SSE standards requires students:
- to write expressions in equivalent forms to reveal key quantities in terms of its context.
- to choose and use appropriate mathematics to analyze situations.
Example: The expression represents the cost in dollars of the materials and labor needed to build a square fence with side length x feet around a playground. Interpret the constants and coefficients of the expression in context.
Example: A rectangle has a length that is 2 units longer than the width. If the width is increased by 4 units and the length increased by 3 units, write two equivalent expressions for the area of the rectangle.
Solution: The area of the rectangle is . Students should recognize as the length of the modified rectangle and as the width. Students can also interpret as the sum of the three areas (a square with side length x, a rectangle with side lengths 9 and x, and another rectangle with area 20 that have the same total area as the modified rectangle.
Example: Given that income from a concert is the price of a ticket times each person in attendance, consider the equation that represents income from a concert where p is the price per ticket. What expression could represent the number of people in attendance?
Solution: The equivalent factored form, , shows that the income can be interpreted as the price times the number of people in attendance based on the price charged. Students recognize as a single quantity for the number of people in attendance.
Example: The expression is the amount of money in an investment account with interest compounded annually for n years. Determine the initial investment and the annual interest rate.
Note: The factor of 1.055 can be rewritten as , revealing the growth rate of 5.5% per year.
Instructional Resources
Tasks / Additional Resources
FAL: Generating Polynomials from Patterns (Math Assessment Project) NEW
Back to: Table of Contents
Algebra – Seeing Structure in Expressions
NC.M1.A-SSE.3Writeexpressionsin equivalent forms tosolveproblems.
Writeanequivalent formof a quadraticexpression byfactoring, whereisaninteger of the quadraticexpression,,torevealthesolutionsoftheequation orthe zeros of the function the expression defines.
Concepts and Skills / The Standards for Mathematical Practices
Pre-requisite / Connections
- Factoring and expanding linear expressions with rational coefficients (7.EE.1)
- Understand that rewriting expressions into equivalent forms can reveal other relationships between quantities (7.EE.2)
4 – Model with mathematics
7 – Look for and make use of structure.
Connections / Disciplinary Literacy
- Interpreting the factors in context (NC.M1.A-SSE.1b)
- Understanding the relationship between factors, solutions, and zeros (NC.M1.A-APR.3)
- Solving quadratic equations (NC.M1.A-REI.4)
- Rewriting quadratic functions into different forms to show key features of the function (NC.M1.F-IF.8a)
Students should be able to compare and contrast the zeros of a function and the solutions of a function.
New Vocabulary: quadratic expression, zeros, linear factors
Mastering the Standard
Comprehending the Standard / Assessing for Understanding
Students factor a quadratic in the form where is an integer in order to reveal the zeroes of the quadratic function.
Students use the linear factors of a quadratic function to explain the meaning of the zeros of quadratic functions and the solutions to quadratic equations in a real-world problem. / Students should understand that the reasoning behind rewriting quadratic expressions into factored form is to reveal different key features of a quadratic function, namely the zeros/x-intercepts.
Example: The expression represents the height of a coconut thrown from a person in a tree to a basket on the ground where x is the number of seconds.
a)Rewrite the expression to reveal the linear factors.
b)Identify the zeroes and intercepts of the expression and interpret what they mean in regard to the context.
c)How long is the ball in the air?
Example: Part A: Three equivalent equations for are shown. Select the form that reveals the zeros of without changing the form of the equation.
Part B: Select all values of for which .
(from the Smarter Balanced Assessment Consortium)
Students should understand that the reasoning behind rewriting quadratic expressions into factored form is to reveal the solutions to quadratic equations.
Example: A vacant rectangular lot is being turned into a community vegetable garden with a uniform path around it. The area of the lot is represented by where is the width of the path in meters. Find the width of the path surrounding the garden.
Instructional Resources
Tasks / Additional Resources
Graphs of Quadratic Functions (Illustrative Mathematics)NEW
Back to: Table of Contents
Algebra – ArithmeticwithPolynomialExpressions
NC.M1.A-APR.1Performarithmeticoperationsonpolynomials.
Build an understanding that operations with polynomials are comparable to operations with integers by adding and subtracting quadratic expressions and byadding, subtracting, and multiplying linear expressions.
Concepts and Skills / The Standards for Mathematical Practices
Pre-requisite / Connections
- Add, subtract, factor and expand linear expressions (7.EE.1)
- Understand that rewriting expressions into equivalent forms can reveal other relationships between quantities (7.EE.2)
2 – Reason abstractly and quantitatively
7 – Look for and make use of structure
Connections / Disciplinary Literacy
- Rewrite expressions using the properties of exponents (NC.M1.N-RN.2)
- Understanding the process of elimination (NC.M1.A-REI.5)
- Rewrite a quadratic function to reveal key features (NC.M1.F-IF.8a)
- Building functions to model a relationship (NC.M1.F-BF.1b)
Students should be able to compare operations with polynomials to operations with integers.
New Vocabulary: polynomial, quadratic expression
Mastering the Standard
Comprehending the Standard / Assessing for Understanding
Students connect their knowledge of integer operations to polynomial operations.
At the Math 1 level, students are only responsible for the following operations:
- adding and subtracting quadratic expressions
- adding, subtracting, and multiplying linear expressions
Example: Write at least two equivalent expressions for the area of the circle with a radius of kilometers.
Example: Simplify each of the following:
a)
b)
Example:The area of a trapezoid is found using the formula , where is the area, is the height, and and are the lengths of the bases.
What is the area of the above trapezoid?
A)
B)
C)
D)
(NCDPI Math I released EOC #33)
Example: A town council plans to build a public parking lot. The outline below represents the proposed shape of the parking lot.
a)Write an expression for the area, in square feet, of this proposed parking lot. Explain the reasoning you used to find the expression.
b)The town council has plans to double the area of the parking lot in a few years. They plan to increase the length of the base of the parking lot by p yards, as shown in the diagram below.
Write an expression in terms of x to represent the value of p, in feet. Explain the reasoning you used to find the value of p.
Example: A cardboard box as a height of x, a width that is 3 units longer than the height, and a length that is 2 units longer than the width. Write an expression in terms of x to represent the volume of the box.
Instructional Resources
Tasks / Additional Resources
FAL: Generating Polynomials from Patterns (Math Assessment Project)NEW
Back to: Table of Contents